My new favourite logic puzzle

By Brian Clegg

I enjoy a good logic puzzle, often those involving the more mind-boggling aspects of probability, such as Monty Hall and the Tuesday’s child puzzles. But I’ve recently come across a more pure logic one that it took me a while to get my head around. It’s apparently an old one, much debated on social media, so apologies if you’ve already come across it.

I came across it while reading to review Steven Pinker’s book When Everyone Knows That Everyone Knows soon out in paperback (my review will be out around 3 September), which has quite a few statements in that take a little getting your head around. Apparently many of the settings of this logic puzzle, involving harems or killing people, are potentially triggering, so Pinker reworked in using a group of psychologists and the problem of spinach on the teeth. 

I’m essentially repeating his version, but with logicians rather than psychologists as I’m not convinced many of the psychologists I’ve met score well on the logical thinking front.

We have a group of a dozen or so logicians, meeting for a regular dinner. After the meal, as they are sipping their ports, the person responsible for calling them together stands up and announces ‘At least one of you has spinach on your teeth. In a moment I shall shout “Reveal!” If you know that you have spinach on your teeth (without checking in a mirror or using your phone), stand up.’

The organiser shouts ‘Reveal!’ No one stands up. Giving them a little time to think of the consequences of what just happened, the organiser shouts ‘Reveal!’ again. No one stands up. Giving them rather longer to think, the organiser shouts ‘Reveal!’ Three people stand up. Why?

If you haven’t come across it already, do try to work out what’s going on – it initially feels impossible that any extra information has been shared, but the important thing to consider is the implications of people standing up each time in turn – they change as a consequence of what can be learned by people not standing up.

… a little longer to think if you aren’t there yet.

Imagine there was just one person with spinach on their teeth. When they looked at everyone else, without any spinach in sight, then they would have stood up on the first shout of ‘Reveal!’ because it had to be them.

As no one stood up, the players now have an extra piece of information: they know that at least two people have spinach on their teeth. If a player saw just one person with spinach they would know that they had it themselves. As they’re logicians, the two people with spinach would work this out, and would both stand up on the second should ‘Reveal!’

But no one stood up. So now the players with spinach on their teeth, who can see two lots of spinach on the others, know that those two people are not the only ones. Realising, then, that they make up the third, all three will stand up on the third call of ‘Reveal!’

Incidentally, I think Pinker’s book is very good, but unfortunately he doesn’t explain this particular puzzle very well – I hope I have done so more effectively. Incidentally, if you like puzzles, do take a look at my book Conundrum. 

Image from Unsplash+ by Ahmet Kurt

See all Brian’s online articles or subscribe to a weekly email free here 

Now Appearing is the blog of science writer Brian Clegg (www.brianclegg.net), author of Inflight Science, Before the Big Bang and The God Effect.

Source: http://brianclegg.blogspot.com/2026/08/my-new-favourite-logic-puzzle.html